Total Energy

About

The total energy of a molecule in quantum mechanics is a comprehensive quantity that accounts for all the interactions between the electrons and nuclei that make up the molecule. In quantum mechanics, the total energy is derived from solving the Schrödinger equation, which describes how the quantum state of a physical system changes with time. For molecules, this involves interactions between the electrons and nuclei, considering both kinetic and potential energies.

The Molecular Schrödinger Equation

The time-independent Schrödinger equation for a molecule is given by:

$\hat{H} \Psi = E_{\text{total}} \Psi$

Where:

• $\hat{H}$ is the Hamiltonian operator (the total energy operator of the system).

• $\Psi$ is the wavefunction of the molecule, describing the quantum state of the system (including both electronic and nuclear degrees of freedom).

• $E_{\text{total}}$ is the total energy (eigenvalue) of the molecule.

Components of the Hamiltonian

The total energy of a molecule comes from various contributions that are incorporated into the Hamiltonian operator $\hat{H}$. The Hamiltonian includes both kinetic and potential energy terms for all particles (electrons and nuclei) in the system.

$\hat{H} = \hat{T}n + \hat{T}e + \hat{V}{nuc-nuc} + \hat{V}{nuc-elec} + \hat{V}_{elec-elec}$

Where:

• $\hat{T}_n$ is the kinetic energy of the nuclei.

• $\hat{T}_e$ is the kinetic energy of the electrons.

• $\hat{V}_{nuc-nuc}$ is the potential energy due to repulsion between nuclei.

• $\hat{V}_{nuc-elec}$ is the potential energy from the attraction between electrons and nuclei.

• $\hat{V}_{elec-elec}$ is the potential energy due to repulsion between electrons.

Breakdown of Total Energy

1. Electronic Energy ($E_{\text{electronic}}$):

   • The electronic energy is the contribution from the kinetic energy of the electrons and the potential energy of their interactions with the nuclei and with each other. It is the most significant part of the total energy and is responsible for determining the chemical bonding and molecular structure.

   • In practice, the electronic part of the total energy is computed using the Born-Oppenheimer approximation, where the nuclei are treated as stationary due to their much larger mass compared to electrons. This decouples the electronic and nuclear motions.

2. Nuclear Repulsion Energy ($E_{\text{nuc-nuc}}$):

   • This is the energy due to the electrostatic repulsion between the positively charged nuclei of the atoms in the molecule.

   • Since nuclei are much heavier than electrons, they are usually considered fixed during electronic energy calculations, and the nuclear repulsion is added as a classical term to the total energy.

3. Vibrational Energy ($E_{\text{vibrational}}$):

   • Molecules are not static but vibrate around their equilibrium positions. The energy associated with these vibrational modes is called vibrational energy.

   • In quantum mechanics, vibrational energy is quantized, and each mode has discrete vibrational levels.

4. Rotational Energy ($E_{\text{rotational}}$):

   • This is the energy associated with the overall rotation of the molecule. Like vibrational energy, rotational energy is quantized.

   • The rotational energy levels are closely spaced compared to vibrational levels, making them important in molecular spectroscopy, particularly at lower temperatures.

5. Translational Energy ($E_{\text{translational}}$):

   • Translational energy is the kinetic energy associated with the movement of the entire molecule through space. It is particularly relevant in gases and is related to the temperature of the system.

   • In quantum mechanics, translational motion is also quantized, but for most practical purposes, it can be treated classically for large molecules.

6. Zero-Point Energy (ZPE):

   • Zero-point energy is the residual vibrational energy that a molecule possesses even at absolute zero temperature. It arises from the fact that vibrational modes never completely stop, due to the Heisenberg uncertainty principle.

   • This is the energy of the molecule in its ground vibrational state.

The Born-Oppenheimer Approximation

Due to the vast difference in masses between electrons and nuclei, molecular total energy calculations often make use of the Born-Oppenheimer approximation. This approximation assumes that the nuclei are fixed in place (because they move much more slowly than electrons), and the electronic Schrödinger equation is solved first.

1. Electronic Total Energy:

   • The total electronic energy, $E_{\text{electronic}}$, is found by solving the Schrödinger equation for the electrons, keeping the nuclear positions fixed. This electronic energy includes both kinetic energy and potential energy due to electron-nucleus and electron-electron interactions.

2. Nuclear Motion:

   • After the electronic energy is determined, the vibrational and rotational energy levels are calculated by considering the motion of the nuclei. These energies are added as corrections to the electronic energy to determine the full total energy of the molecule.

Total Energy Expression

The total energy of a molecule, including all components, can be expressed as:

$E_{\text{total}} = E_{\text{electronic}} + E_{\text{nuc-nuc}} + E_{\text{vibrational}} + E_{\text{rotational}} + E_{\text{translational}} + ZPE$

Where:

• $E_{\text{electronic}}$ is the energy due to electronic interactions.

• $E_{\text{nuc-nuc}}$ is the nuclear repulsion energy.

• $E_{\text{vibrational}}$, $E_{\text{rotational}}$, and $E_{\text{translational}}$ are the quantized energies due to vibrational, rotational, and translational motion, respectively.

• ZPE is the zero-point vibrational energy.

Significance of Total Energy in Quantum Mechanics

1. Molecular Stability:

   • The total energy provides insights into the stability of a molecule. Lower total energy generally indicates a more stable molecule.

   • The energy difference between the reactants and products in a chemical reaction is related to the reaction enthalpy and tells whether the reaction is exothermic or endothermic.

2. Reaction Energies:

   • The difference in total energy between reactants, transition states, and products allows chemists to predict reaction mechanisms, activation energies, and rate constants.

3. Spectroscopy:

   • Differences in total energy between quantum states of a molecule correspond to the absorption or emission of photons, making the total energy crucial for interpreting spectroscopic data (e.g., IR, UV-Vis, NMR).

4. Thermodynamics:

   • From the total energy, one can derive other thermodynamic quantities such as enthalpy, free energy, and entropy, which are important for understanding chemical equilibria and phase transitions.

Summary

In quantum mechanics, the total energy of a molecule is the sum of various contributions, including electronic, vibrational, rotational, and translational energies, along with nuclear repulsion and zero-point energy. It is determined by solving the Schrödinger equation for the molecular system, often using the Born-Oppenheimer approximation. This energy provides critical insights into the molecular structure, stability, chemical reactivity, and spectroscopic properties of the molecule.

Method

The Total Energy was calculated using GFN2-xTB within xTB 6.6.0

Find

The Total Energy is found in the Global property table: